Heffernan sustained the injury early in the 0-0 draw with St Mirren and boss Terry Butcher confirmed the striker, signed from Kilmarnock in the summer, is set for a spell on the sidelines.
Butcher told the club's website: "Paul had a scan last night (Monday) and we're awaiting the report from that.
"But we probably think he'll be out for somewhere between three to six weeks. It was a bit of a hammer blow coming so early into the match - just five minutes into the game at St Mirren.
"So it's unlucky for Paul, but it gives James Collins a great chance to get into the team. When one door closes, another one opens."